Category Archives: research

I recently finished reading a series of excellent articles by Peter Holy, Regula Krapf, Philipp Lücke, Ana Njegomir and Philipp Schlicht investigating properties of class forcing over models of ${\rm GBC}$ (Gödel-Bernays set theory). So I would like to summarize … Continue reading →

In the previous post, I discussed a theorem of Woodin, extended recently by Blanck and Enayat, showing that for every computably enumerable theory $T$, there is in index $e$ such that if $M\models T$ satisfies that $W_e$ is contained in … Continue reading →

My former PhD supervisor, collaborator, and dear friend Joel David Hamkins is turning 50 this year. Evidently, Joel is much too young for a full-blown birthday conference, but some celebration must still be had. So to celebrate Joel’s birthday, we … Continue reading →

Last Saturday, I participated in my first Black Board Day, an informal annual workshop dedicated to Gödel and organized for the past ten years by the neuroscientist Memming Park and his enthusiastic group of scientist friends.

This post is motivated by a really great paper of Wojciech Guzicki from the 1970s entitled “On weaker forms of choice in second-order arithmetic” [1] that I recently stumbled on while trying to trace the history of choice principles in … Continue reading →

I was lucky enough to start graduate school and end up choosing the same adviser with two people who still remain my favorite colleagues and dearest friends. Coincidentally, the adviser also turned out to be quite good. The students were … Continue reading →

If $W$ is a transitive set or class and it thinks that there is an elementary embedding $j$ between some first-order structures $\mathcal M$ and $\mathcal N$, then $j$ is an actual elementary embedding and so the universe $V$ agrees … Continue reading →

Joel Hamkins recently wrote an excellent post on Kelley-Morse set theory (${\rm KM}$) right here on Boolesrings. I commented on the post about the variations one finds of what precisely is included in the ${\rm KM}$ axioms. I claimed that … Continue reading →

Ali, I am just seeing the comment because I have been away from MO for a while. So you are saying that even an ill-founded ultrapower by a non-normal filter as long as it is weakly amenable can be used to show that kappa is weakly compact in M?

Let's make some additional assumptions on the ultrafilter $U$. Suppose $M\models{\rm ZFC}^-$ and $\kappa$ is a cardinal in $M$. We say that $U$ is an $M$-ultrafilter if $\langle M,\in,U\rangle$ satisfies that $U$ is a $\kappa$-complete normal ultrafilter on $\kappa$. Because $U$ is only $\kappa$-complete for sequences in $M$ and $M$ might be missing even countable […]

← Older revision Revision as of 11:23, 15 March 2018 Line 3: Line 3: For those interested, there is a discord server created by me and [[User:Zetapology|Zetapology]]: https://discord.gg/tbGrU39 For those interested, there is a discord server created by me and [[User:Zetapology|Zetapology]]: https://discord.gg/tbGrU39 −My "ideal background theory" (or just "prefered theory") is $\text{ZFC + GCH + […]